Coarsely quantized MIMO signalling methods have gained popularity in the recent developments of massive MIMO as they open up opportunities for massive MIMO implementation using cheap and power-efficient radio-frequency front-ends. This paper presents a new one-bit MIMO precoding approach using spatial Sigma-Delta ($ΣΔ$) modulation. In previous one-bit MIMO precoding research, one mainly focuses on using optimization to tackle the difficult binary signal optimization problem that arises from the precoding design. Our approach attempts a different route. Assuming angular MIMO channels, we apply $ΣΔ$ modulation---a classical concept in analog-to-digital conversion of temporal signals---in space. The resulting $ΣΔ$ precoding approach has two main advantages: First, we no longer need to deal with binary optimization in $ΣΔ$ precoding design. Particularly, the binary signal restriction is replaced by peak signal amplitude constraints. Second, the impact of the quantization error can be well controlled via modulator design and under appropriate operating conditions. Through symbol error probability analysis, we reveal that the very large number of antennas in massive MIMO provides favorable operating conditions for $ΣΔ$ precoding. In addition, we develop a new $ΣΔ$ modulation architecture that is capable of adapting the channel to achieve nearly zero quantization error for a targeted user. Furthermore, we consider multi-user $ΣΔ$ precoding using the zero-forcing and symbol-level precoding schemes. These two $ΣΔ$ precoding schemes perform considerably better than their direct one-bit quantized counterparts, as simulation results show.